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BK

Second edition

Weinheim : Wiley-VCH, [2020]

xxi, 921 stran : ilustrace ; 25 cm

ISBN 978-3-527-34553-3 (vázáno)

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This textbook introduces the fundamental concepts of quantum mechanics such as waves, particles and probability before explaining the postulates of quantum mechanics in detail. In the proven didactic manner, the textbook then covers the classical scope of introductory quantum mechanics, namely simple two-level systems, the one-dimensional harmonic oscillator, the quantized angular momentum and particles in a central potential. The entire book has been revised to take into account new developments in quantum mechanics curricula. The textbook retains its original style in this new edition: fundamental concepts are explained in chapters, each followed by complements that provide more detailed discussions, examples and applications..

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Volume I // Table of contents vii // I WAVES AND PARTICLES. INTRODUCTION TO THE BASIC // IDEAS OF QUANTUM MECHANICS 1 // A Electromagnetic waves and photons 3 // B Material particles and matter waves 10 // C Quantum description of a particle. Wave packets 13 // D Particle in a time-independent scalar potential 23 // READER’S GUIDE FOR COMPLEMENTS 33 // AI Order of magnitude of the wavelengths associated with material // particles 35 // BI Constraints imposed by the uncertainty relations 39 // 1 Macroscopic system 39 // 2 Microscopic system 40 // CI Heisenberg relation and atomic parameters 41 // DI An experiment illustrating the Heisenberg relations 45 // EI A simple treatment of a two-dimensional wave packet 49 // 1 Introduction 49 // 2 Angular dispersion and lateral dimensions 49 // 3 Discussion 51 // FI The relationship between one- and three-dimensional problems 53 // 1 Three-dimensional wave packet 53 // 2 Justification of one-dimensional models 56 // GI One-dimensional Gaussian wave packet: spreading of the wave packet 57 // 1 Definition of a Gaussian wave packet 57 // 2 Calculation of Ax and Ap; uncertainty relation 58 // 3 Evolution of the wave packet 59 // HI Stationary states of a particle in one-dimensional square potentials 63 // 1 Behavior of a stationary wave function ip(x) 63 // 2 Some simple cases 65 // JI Behavior of a wave packet at a potential step 75 // 1 Total reflection: E < V0 75 // 2 Partial reflection: E > V0 79 // KI Exercises 83 // II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS 87 // A Space of the one-particle wave function 88 // ? State space. Dirac notation 102 // C Representations in state space 116 // D Eigenvalue equations. Observables 126 // E Two important examples of representations and observables 139 // F Tensor product of state spaces 147 // READER’S GUIDE FOR COMPLEMENTS 159 // The Schwarz inequality 161 //

BII Review of some useful properties of linear operators 163 // 1 Trace of an operator 163 // 2 Commutator algebra 165 // 3 Restriction of an operator to a subspace 165 // 4 Functions of operators 166 // 5 Derivative of an operator 169 // ?? Unitary operators 173 // 1 General properties of unitary operators 173 // 2 Unitary transformations of operators 177 // 3 The infinitesimal unitary operator 178 // Dn A more detailed study of the r and p representations 181 // 1 The (|r)} representation 181 // 2 The (|p)} representation 184 // ?? Some general properties of two observables, Q and P, whose commutator is equal to ih 187 // 1 The operator 5(A): definition, properties 187 // 2 Eigenvalues and eigenvectors of Q 188 // 3 The {|q)} representation 189 // 4 The (|p)} representation. The symmetric nature of the P and Q observables 190 // ?? The parity operator 193 // 1 The parity operator 193 // 2 Even and odd operators 196 // 3 Eigenstates of an even observable B+ 199 // 4 Application to an important special case 199 // ?? An application of the properties of the tensor product: the twodimensional infinite well 201 // 1 Definition; eigenstates 201 // 2 Study of the energy levels 202 // Exercises 205 // III THE POSTULATES OF QUANTUM MECHANICS 213 // A Introduction 214 // B Statement of the postulates 215 // C The physical interpretation of the postulates concerning observables and their measurement 226 // D The physical implications of the Schrödinger equation 237 // E The superposition principle and physical predictions 253 // READER’S GUIDE FOR COMPLEMENTS 267 // Am Particle in an infinite one-dimensional potential well 271 // 1 Distribution of the momentum values in a stationary state 271 // 2 Evolution of the particle’s wave function 275 // 3 Perturbation created by a position measurement 279 // ?? Study of the probability current in some special cases 283 //

1 Expression for the current in constant potential regions 283 // 2 Application to potential step problems 284 // 3 Probability current of incident and evanescent waves, in the case of reflection // from a two-dimensional potential step 285 // Cm Root mean square deviations of two conjugate observables 289 // 1 The Heisenberg relation for P and Q 289 // 2 The “minimum” wave packet 290 // Dm Measurements bearing on only one part of a physical system 293 // 1 Calculation of the physical predictions 293 // 2 Physical meaning of a tensor product state 295 // 3 Physical meaning of a state that is not a tensor product 296 // Em The density operator 299 // 1 Outline of the problem 299 // 2 The concept of a statistical mixture of states 299 // 3 The pure case. Introduction of the density operator 301 // 4 A statistical mixture of states (non-pure case) 304 // 5 Use of the density operator: some applications 308 // FIII The evolution operator 313 // 1 General properties 313 // 2 Case of conservative systems 315 // Gm The Schrödinger and Heisenberg pictures 317 // HIII Gauge invariance 321 // 1 Outline of the problem: scalar and vector potentials associated with an electromagnetic field; concept of a gauge 321 // 2 Gauge invariance in classical mechanics 322 // 3 Gauge invariance in quantum mechanics 327 // JIII Propagator for the Schrödinger equation 335 // 1 Introduction 335 // 2 Existence and properties of a propagator if (2,1) 336 // 3 Lagrangian formulation of quantum mechanics 339 // KIII Unstable states. Lifetime 343 // 1 Introduction 343 // 2 Definition of the lifetime 344 // 3 Phenomenological description of the instability of a state 345 // Exercises 347 // MIII Bound states in a “potential well” of arbitrary shape 359 // 1 Quantization of the bound state energies 359 // 2 Minimum value of the ground state energy 363 //

NIII Unbound states of a particle in the presence of a potential well or barrier 367 // 1 Transmission matrix M(k) 368 // 2 Transmission and reflection coefficients 372 // 3 Example 373 // Quantum properties of a particle in a one-dimensional periodic structure 375 // 1 Passage through several successive identical potential barriers 376 // 2 Discussion: the concept of an allowed or forbidden energy band 381 // 3 Quantization of energy levels in a periodic potential; effect of boundary conditions 383 // IV APPLICATIONS OF THE POSTULATES TO SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMS 393 // A Spin 1/2 particle: quantization of the angular momentum 394 // B Illustration of the postulates in the case of a spin 1/2 401 // C General study of two-level systems 411 // READER’S GUIDE FOR COMPLEMENTS 423 // Aiv The Pauli matrices 425 // 1 Definition; eigenvalues and eigenvectors 425 // 2 Simple properties 426 // 3 A convenient basis of the 2x2 matrix space 427 // Biv Diagonalization of a 2 x 2 Hermitian matrix 429 // 1 Introduction 429 // 2 Changing the eigenvalue origin 429 // 3 Calculation of the eigenvalues and eigenvectors 430 // Civ Fictitious spin 1/2 associated with a two-level system 435 // 1 Introduction r 435 // 2 Interpretation of the Hamiltonian in terms of fictitious spin 435 // 3 Geometrical interpretation 437 // Div System of two spin 1/2 particles 441 // 1 Quantum mechanical description 441 // 2 Prediction of the measurement results 444 // Eiv Spin 1/2 density matrix 449 // 1 Introduction 449 // 2 Density matrix of a perfectly polarized spin (pure case) 449 // 3 Example of a statistical mixture: unpolarized spin 450 // 4 Spin 1/2 at thermodynamic equilibrium in a static field 452 // 5 Expansion of the density matrix in terms of the Pauli matrices 453 // Fjv Spin 1/2 particle in a static and a rotating magnetic fields: magnetic resonance 455 //

1 Classical treatment; rotating reference frame 455 // 2 Quantum mechanical treatment 458 // 3 Relation between the classical treatment and the quantum mechanical treatment: evolution of (M) 463 // 4 Bloch equations 463 // Giv A simple model of the ammonia molecule 469 // 1 Description of the model 469 // 2 Eigenfunctions and eigenvalues of the Hamiltonian 471 // 3 The ammonia molecule considered as a two-level system 477 // Hiv Effects of a coupling between a stable state and an unstable state 485 // 1 Introduction. Notation 485 // 2 Influence of a weak coupling on states of different energies 486 // 3 Influence of an arbitrary coupling on states of the same energy 487 // Jiv Exercises 491 // // V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR 497 // A Introduction 497 // B Eigenvalues of the Hamiltonian 503 // C Eigenstates of the Hamiltonian 510 // D Discussion 518 // READER’S GUIDE FOR COMPLEMENTS // 525 // Av Some examples of harmonic oscillators 527 // 1 Vibration of the nuclei of a diatomic molecule 527 // 2 Vibration of the nuclei in a crystal 534 // 3 Torsional oscillations of a molecule: ethylene 536 // 4 Heavy muonic atoms 541 // Bv Study of the stationary states in the x representation. Hermite polynomials 547 // 1 Hermite polynomials 547 // 2 The eigenfunctions of the harmonic oscillator Hamiltonian 550 // Cv Solving the eigenvalue equation of the harmonic oscillator by the polynomial method 555 // 1 Changing the function and the variable 555 // 2 The polynomial method 557 // Dv Study of the stationary states in the momentum representation 563 // 1 Wave functions in momentum space 563 // 2 Discussion 565 // Ev The isotropic three-dimensional harmonic oscillator 569 // 1 The Hamiltonian operator 569 // 2 Separation of the variables in Cartesian coordinates 570 // 3 Degeneracy of the energy levels 572 //

Fv A charged harmonic oscillator in a uniform electric field 575 // 1 Eigenvalue equation of H\\S) in the (|x)} representation 575 // 2 Discussion 577 // 3 Use of the translation operator 579 // Gy Coherent “quasi-classical” states of the harmonic oscillator 583 // 1 Quasi-classical states 584 // 2 Properties of the |a) states 588 // 3 Time evolution of a quasi-classical state 594 // 4 Example: quantum mechanical treatment of a macroscopic oscillator 596 // Hv Normal vibrational modes of two coupled harmonic oscillators 599 // 1 Vibration of the two coupled in classical mechanics 599 // 2 Vibrational states of the system in quantum mechanics 605 // Jv Vibrational modes of an infinite linear chain of coupled harmonic oscillators; phonons 611 // 1 Classical treatment 612 // 2 Quantum mechanical treatment 622 // 3 Application to the study of crystal vibrations: phonons 626 // Kv Vibrational modes of a continuous physical system. Photons 631 // 1 Outline of the problem 631 // 2 Vibrational modes of a continuous mechanical system: example of a vibrating // string 632 // 3 Vibrational modes of radiation: photons 639 // Lv One-dimensional harmonic oscillator in thermodynamic equilibrium // at a temperature T 647 // 1 Mean value of the energy 648 // 2 Discussion 650 // 3 Applications 651 // 4 Probability distribution of the observable X 655 // My Exercises 661 // VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS 667 // A Introduction: the importance of angular momentum 667 // B Commutation relations characteristic of angular momentum 669 // C General theory of angular momentum 671 // D Application to orbital angular momentum 685 // READER’S GUIDE FOR COMPLEMENTS 703 // Avi Spherical harmonics 705 // 1 Calculation of spherical harmonics 705 // 2 Properties of spherical harmonics 710 // B vi Angular momentum and rotations 717 // 1 Introduction 717 //

2 Brief study of geometrical rotations & 718 // 3 Rotation operators in state space. Example: a spinless particle 720 // 4 Rotation operators in the state space of an arbitrary system 727 // 5 Rotation of observables 730 // 6 Rotation invariance 734 // Cvi Rotation of diatomic molecules 739 // 1 Introduction 739 // 2 Rigid rotator. Classical study 740 // 3 Quantization of the rigid rotator 741 // 4 Experimental evidence for the rotation of molecules 746 // Dvi Angular momentum of stationary states of a two-dimensional harmonic oscillator 755 // 1 Introduction 755 // 2 Classification of the stationary states by the quantum numbers nx and ny 759 // 3 Classification of the stationary states in terms of their angular momenta 761 // 4 Quasi-classical states // 765 // Evi A charged particle in a magnetic field: Landau levels 771 // 1 Review of the classical problem 771 // 2 General quantum mechanical properties of a particle in a magnetic field 775 // 3 Case of a uniform magnetic field 779 // Fvi Exercises 795 // VII PARTICLE IN A CENTRAL POTENTIAL, HYDROGEN ATOM 803 // A Stationary states of a particle in a central potential 804 // B Motion of the center of mass and relative motion for a system of two interacting particles 812 // C The hydrogen atom 818 // READER’S GUIDE FOR COMPLEMENTS 831 // Avii Hydrogen-like systems 833 // 1 Hydrogen-like systems with one electron 834 // 2 Hydrogen-like systems without an electron 839 // Bvii A soluble example of a central potential: The isotropic three-dimensional harmonic oscillator 841 // 1 Solving the radial equation 842 // 2 Energy levels and stationary wave functions 845 // Cvii Probability currents associated with the stationary states of the hydrogen atom 851 // 1 General expression for the probability current 851 // 2 Application to the stationary states of the hydrogen atom 852 //

Dvii The hydrogen atom placed in a uniform magnetic field. Paramagnetism and diamagnetism. The Zeeman effect 855 // 1 The Hamiltonian of the problem. The paramagnetic term and the diamagnetic term 856 // 2 The Zeeman effect 862 // Evii Some atomic orbitals. Hybrid orbitals 869 // 1 Introduction 869 // 2 Atomic orbitals associated with real wave functions 870 // 3 sp hybridization 876 // 4 sp2 hybridization 878 // 5 sp3 hybridization 882 // Fvii Vibrational-rotational levels of diatomic molecules 885 // 1 Introduction r 885 // 2 Approximate solution of the radial equation 886 // 3 Evaluation of some corrections 892 // Gvn Exercises 899 // 1 Particle in a cylindrically symmetric potential 899 // 2 Three-dimensional harmonic oscillator in a uniform magnetic field 899 // INDEX 901